Active Thermal Insulation of High-Temperature Surfaces of Technological Equipment Based on Packed Beds

Models of heat transfer in infiltrated packed beds of large low-heat conducting particles that are used as active thermal insulation of high-temperatures surfaces of technological equipment have been developed. A criterion of active thermal insulation involving main geometrical and thermal parameters of the infiltrated packed bed has been introduced. The effect of packed bed nonisothermicity and gas compressibility on the resistance of active thermal insulation has been investigated. An approximation dependence of the resistance of the bed on its geometrical, hydrodynamic, and thermal characteristics has been obtained. Cooling of a cap gas distributer via active insulation has been studied.


Introduction
Thermal insulation made of low-heat conducting packed material is widely used in practice.Infiltration of thermal insulation, as in the case of porous highly heat-conducting matrices of heat protecting coatings (Polyaev, Maiorov, & Vasil'ev, 1988), substantially increases the intensity of heat and mass transfer processes and, thus, the efficiency of thermal insulation.In a number of industrial objects, e.g., heating furnaces, moderate-and small-power boilers, the efficiency of such thermal insulation that can be called active increases due to the fact that heated heat carrier (e.g., air) can be used in a technological process, thus improving its efficiency (heating of air before its supply to the furnace etc.).In such cases thermal insulation also plays a role of air heater.

Main Schemes of Heat Carrier Motion in Active Thermal Insulation
Thermal insulation in the form of an annular packed bed falls well in the arrangement of a cylindrical high-temperature chamber, e.g., cyclone-bed furnace.Figure 1 presents the main schemes of heat carrier motion, viz., radial (Figure 1a), longitudinal (Figure 1b), and tangential (Figure 1c). Figure 1.Main schemes of heat carrier motion in active thermal insulation: 1) combustion chamber; 2) lining of the combustion chamber; 3) packed bed; 4) channel of heated air; 5) channel of cold air; 6) tangential nozzles

Scheme of Figure 1a
Problem formulation.Taking into account a relatively small curvature of the furnace surface, in order to describe the process we use the two-temperature model for a plane bed where radiation heat transfer through the particle skeleton is disregarded: with the boundary conditions (5) The model parameters α, α 0 , λ f , λ s , and σ r are calculated by the relations (Frank-Kamenetskii, 1967;Polezhaev, & Selivestrov, 2002;Aerov, Todes, & Narinskii, 1979;Teplitskii, & Kovenskii, 2011;Borodulya, & Kovenskii, 1983): ( ) It should be noted that the second term in the denominator of (6) takes into account the effect of thermal resistance of a particle on interphase heat transfer (Frank-Kamenetskii, 1967).
To solve the problem posed we use the equation of the total heat balance We determine the dimensionless relative temperatures of the phases in terms of the gas temperature at the outlet from the bed The initial problem (1)−( 5) is solved at two stages: the first stage is the determination of relative temperatures θ s (ξ) and θ f (ξ); the second stage is the determination of the absolute temperatures of gas T f and particles T s .In the dimensionless form the system of equations (1)−( 5) for finding θ s (ξ) and θ f (ξ) is written with account for (13) as ( ) As is seen, the use of the balance relation ( 13) and determination of dimensionless temperatures in terms of T f (δ) in the form of ( 14) allows substantial simplification of the form of the boundary condition for particles at the outlet from the bed.It should be noted that usually in modeling heat-protection elements it is assumed that λ f =0 and conductive heat transfer is taken into account only in a porous highly heat conducting skeleton (Polyaev, Maiorov, & Vasil'ev, 1988).In our case, the skeleton thermal conductivity is small and the convective component λ f plays the main role (see ( 8)).
With due regard for the fact that the Pe f , Pe α , and St 0 numbers are the functions of Re, Pr, and d⁄δ, the Pe s number is a function of Re, Pr, d⁄δ, and 0 c f λ λ , relative temperatures of the phases can be presented in the form of the following relations: Solutions ( 15)−(19) for 0 0 44 that are obtained based on the computer system of mathematical simulation Maple7 are given in Figure 2. In order to implement the problem formulated in the present work and to determine relations for calculation of absolute temperatures of the gas and particles at the outlet from the bed we should find the approximation of the dependence for calculation of relative temperatures of the gas and particles at the outlet from the bed.With account for slight variation of the Pr number within the temperature range 20−1000°C, the functional relation θ s (1) has the form ( ) The obtained approximation of the function f is 0.58 0.75 0 ) In order to find the absolute quantities T f and T s we use condition (13) whose dimensionless form is as follows: The relation between ( ) ( ) With account for (25) expression (24) will take the form of the fourth-power equation relative to ( ) ( ) ( ) ( ) ( ) Bo θ ≤ ≤ with an error not higher than 2%.With account for ( 25) and ( 27) for ( ) we have the formula where ( )  15)-( 19) and ( 26) with account for (25).As is seen, the temperatures ( ) , even in the case of highly heat-conducting material (Figure 4c), slightly differ from unity, which indicates practical equality of T 0 to the temperature of the outer surface of thermal insulation.The same figure shows the profiles of the temperature constructed by the dependence obtained as a result of a similar solution for a one-temperature model (for which ( ) , As is seen, the temperatures are rather close to each other every where except for the outlet zone.This is explained, on the one hand, by the proximity of the equations for f θ ′ and θ ′ , and, on the other hand, by the difference of the boundary conditions ( 5) and (32).

Schemes of Figures 1b and 1c
Problem formulation.Taking into account a relatively small curvature of the furnace surface, we use, for description of the steady-state heat transfer in the lining and the packed bed, a simplified model of plane beds and the corresponding system of equations: with the boundary conditions ) The model parameters are (Aerov, Todes, & Narinskii, 1979;Teplitskii, & Kovenskii, 2011) The heat transfer coefficient α ∞ is determined by the standard method of thermal calculations of boilers (Anonymous, 1998).The quantities λ с and σ r are presented by ( 10)−( 12).
Temperature fields in the lining and the packed bed were calculated by Fluent.An example of solution of the thermal problem for thermal insulation of the boiler with a power of 0.5 MW is shown in Figure 5a.As is seen, the air filtration through the insulation layer leads to the existence of a rather vast area of low temperatures near its outer surface.This indicates that an infiltrated packed bed, though being rather thin (δ = 0.085 m), exhibits good thermo-insulating properties.
Figure 5b show similar results obtained from the problem solution without account for the radiation flux in (36).As is seen, the difference between the results presented in Figure 5a and 5b is not substantial, which is likely related to the effect of large thermal resistance of the lining layer with δ m = 0.11 m on this flow.The result obtained formed the basis for more thorough analysis of thermal mechanics of active thermal insulation with the effect of radiation on it being ignored.In order to obtain an analytical solution convenient for analysis, we considered a simplified version of (33)−( 39) where a one-zone problem of heat conduction at large Peclet numbers 10 = ≥ Solution ( 43)−( 46) is the known solution (Luikov, 1967) of nonstationary problem of plate heating at the boundary conditions of III kind: The expressions for A n and μ n are given in Luikov (1967).
The number includes the main geometric, hydrodynamic, and thermal parameters of the infiltrated packed bed.By virtue of this, the Pe act number may be termed the criterion of active thermal insulation.
Figures 6 and 7 show the temperature distributions over the thickness of thermal insulation at different values of η/Pe act (different lengths of a packed bed).As is seen, at Bi ≥ 5 the dependence of temperature fields on Bi becomes weaker.This also refers to the temperatures at the outer boundary of the bed which determine thermo-insulating properties of the system.The following generalized dependences are obtained for their calculation on the basis of (47): 1.8 We consider another important property of active thermal insulation, i.e., heating of a heat carrier.To calculate it, we determine, based on (47), the temperature that is mean over the cross section y = const: is given in Figure8.A simple analytical presentation of these results is obtained in the form of generalized dependences similar to ( 48) and ( 49): 0.9 0.66 , 1 0.65
For the quantities Bi and Pe act we obtained: Bi = 0.58; Pe act = 3.33.The results of calculation of the temperature distribution on the outer boundary of thermal insulation T(δ, y) and the mean temperature of heat carrier along the length of the packed bed ( ) 〈 〉 T y by ( 48) and ( 51) are listed in Table 1.As is seen, the temperature of the outer side of thermal insulation is low and changes within the range 20-60°C.At the same time, heating of the heat carrier (air) at the outlet from the bed (at η = 1) is substantial and amount to 123°C.

Thermal Conditions of the Cap Gas Distributer With Active Thermal Insulation
Problem formulation.We study thermal conditions of a cap gas distributer with active thermal insulation in the fluidized bed furnace (Figure 9a).As is known (Kunii, & Levenspiel, 1976), this type of gas distributers is widely used in organization of different technological processes in large-scale equipment with fluidized bed.In this case, the role of active insulation is to cool the upper gas-tight wall of lattice 4 and to cool primary air coming into furnace 1 through caps 3.
Figure 9b shows a typical part of the gas distributer and a qualitative character of air streamlines filtered through a packed bed.As is seen, the flow patters is rather complex which makes simulation of the process of convective heat transfer difficult.In this connection, a simplified thermal scheme that takes into account radiative heat transfer and conductive-convective heat exchange between the gas-tight plate of the gas distributer and the packed bed was adopted.
a) b) Figure 9. a) A cap gas distributer with active thermal insulation in a fluidized bed: 1) fluidized bed; 2) furnace lining; 3) cap; 4) upper gas-tight plate of the distributer; 5) packed material; 6) lower perforated wall; 7) subdistributer chamber; 8) discharge branch pipe; 9) air supply; b) part of a gas distributer with air streamlines The main assumptions are as follows: − a one-dimensional model of heat conduction of the infiltrated packed bed of thermal insulation is considered; − the difference between the temperatures of the packed bed phases is neglected which is justifiable at rather high air velocities; − the effect of the bending of air streamlines near the gas-tight wall of the gas distributer (Figure 9b) is taken into account proceeding from the formulation of the regularities of conductive-convective heat transfer in this region based on the wall coefficient of heat transfer αw that is introduced under the conditions of heat carrier motion along the heat transfer surface (Teplitskii, & Kovenskii, 2011;Dekhtyar, Sikovskii, Gorin, & Mukhin, 2002) (Equation ( 42)).
The system of equation that describes air heating in filtration through the packed bed of thermal insulation with a thickness δ has the form The parameters of model ( 53)−( 55) are found in (Aerov, Todes, & Narinskii, 1979;Teplitskii, & Kovenskii, 2011): The characteristic temperature * T involved in ( 57) was taken equal to (T(δ)+T 0 )/2, when λ c is determined by ( 10), and to * w T T = when α w is determined by ( 42).
We write Equations ( 53)−( 55) in the dimensionless form: A specific character of this formulation is that the temperature T w , involved in the boundary condition ( 55) or ( 60) at x = δ is the unknown quantity (only the fluidized bed temperature T b is known a priori, Figure 9b).By virtue of this, in order to calculate the thermal state of a gas distributer with a bed of active thermal insulation one should use the method of successive approximations.
Solution (58)−( 60) has the form ( ) where the coefficient С is determined from the solution of the fourth-power equation which follows from (61): Since the temperature of the air discharging from the packed bed Т(δ) that specifies heating of the air injected into the furnace is the main quantity to be determined, it is more convenient to write (62) in an equivalent form: The quantity T(δ) can be found independently from the balance equation that determines the density of the heat flux through the gas distributer q: ( ) The dimensionless form of ( 64) is Calculation of ( ) 65) is given in Figure 10 for the specified values of w θ ′ = 2, 3, and 4. We took into account that the Bo and St w numbers are not independent and both are determined by the air mass flow rate J f .As is seen, at elevated values of Bo r (elevated air flow rates J f ) Т(δ) begins to change rather weakly.This is, likely, the consequence of the existence of two tendencies.On the one hand, as J f increases, the air passes through a packed bed quicker and thus is less heated.On the other hand, as T(δ) decreases, the heat flux from the fluidized bed q (the right-hand side of ( 65)) increases, thus leading to an increase in T(δ).
In order to calculate the acting temperature mode of the system 'active thermal insulation-upper gas-tight wall of the lattice' at the given temperature of the fluidized bed T b we used the method of successive approximations.First we write the expression for T b on the basis of the equation for the heat-flux density with account for (65) The temperature * w T (Figure 9b) is determined by a similar formula The coefficient of heat exchange between the fluidized bed and the gas-tight surface of the lattice α b is, as is known (Kunii, & Levenspiel, 1976;Baskakov, Matsnev, & Raspopov, 1996), a rather complex function of a large number of parameters of the fluidized bed itself and of the heat transfer surface.But, as is shown in (Baskakov, Matsnev, & Raspopov, 1996), on operation of furnaces under the conditions of developed fluidization of large particles (d b = 1-3 mm) at T b = 850°C with an error not higher than 10% it is assumed to take α b constant and equal to 250 W/(m 2 K).
The schematic diagram of the calculation of temperatures T(δ), T w , and * w T is given in Figure 11.67), does not exceed several dozens of degrees.Heating of air discharging from the packed bed greatly depends on the value of J f and amounts to 30−300 о C. In Figure 13, the profile of temperatures in the insulation which is built by Equation ( 63) for different values of J f is shown.As is seen, the bed temperature begins to increase substantially at ξ > 0.7.1a and 9 The resistance of the infiltrated nonisothermal packed bed is calculated based on the solution of the Navier-Stokes equation for the heat carrier rate V (Goldshtik, 1984)

Schemes of Figures
The coefficient β in the expression for the resistance force is taken in the form ( ) which in the isothermal case corresponds to the Ergun formula (Ergun, 1952).
Representing the resistance force in the form of the Ergun formula, the dependence of the dynamic viscosity of heat carrier and its density on temperature and pressure in the form Integrating ( 70) with respect to ξ going from 0 to 1, we obtain the formula for calculation of the bed resistance: where As follows from (71), it allows for gas compressibility and bed nonisothermicity.We consider particular case of (71): a) An isothermal bed (I 0 = 0; I 1 = I 2 = 1): Equation (71) takes the form ( ) b) Large values of Eu and small thicknesses of the bed δ ; Equation ( 72) is It is evident that (73) is the dimensionless presentation of the classical Ergun formula (Ergun, 1952): ( ) which, probably, does not takes into account the gas compressibility.

Schemes of Figures 1b and 1c
In this case, dependence (71) can also be applied if δ is replaced by L, but the functions I 0 , I 1 , and I 2 are determined by the formulas where the function ( ) , θ ξ η is defined in (47).

27 )Figure 3 .
Figure 3. Dimensionless temperature of particles θ ′ s (1) as a function of the complex Bo r /θ s (1) for different values of T w

Figure 7 .
Figure 7. Distribution of the dimensionless temperature over the bed thickness at small values of

Figure 8 .
Figure 8. Distribution of the mean dimensionless temperature of the bed along its length at different values of the Biot number

Figure 11 .T
Figure 11.Schematic diagram of the calculation of temperatures T(δ), T w , and * w T

Figure
Figure 14 presents the characteristic dependences ( ) 0 act T p Pe p Δ Δ

Table 1 .
The temperature distribution in thermal insulation